Introduction to Large Deviation

2026/2027

Учебный год

ENG

Обучение ведется на английском языке

3

Кредиты

Кто читает:

Факультет математики

Статус:

Дисциплина общефакультетского пула

Когда читается:

2 модуль

Course Syllabus

Abstract

The theory of large deviations underwent a spectacular development in the middle of the last century, driven by various independent groups of researchers. In particular, Donsker and Varadhan studied the large-time asymptotics of empirical measures for Markov processes, while Freidlin and Wentzell investigated the vanishing noise limit of stationary distributions for diffusion processes.

Learning Objectives

• The aim of this course is to give an elementary introduction to the theory of large deviations and to show how it can be applied to study qualitative properties of discrete-time dynamical systems with small noise. In particular, we shall prove convergence of trajectories under the vanishing noise limit and study their deviations from their limits. In the case when the deterministic system has a globally stable equilibrium, we show that the stationary distributions satisfy the large deviation principle and describe the associated rate function.

Expected Learning Outcomes

• The ability to formulate and prove large deviation principles for various stochastic systems by determining their rate functions using Cramér's theorem, Varadhan's lemma, and the contraction principle, as well as to apply these methods to analyze rare event probabilities in statistical mechanics and information theory.

Course Contents

• Laplace asymptotics of integrals
• Weak convergence of probability measures.
• Definition and elementary properties of large deviations
• Gaussian measures and their large deviations
• Discrete-time dynamical systems. Global stability
• Small-noise limits of trajectories with deterministic initial conditions
• Stationary distributions
• Large deviations for stationary distributions of dynamical systems with small noise

Assessment Elements

• homework, active participation
• exam

Interim Assessment

• 2026/2027 2nd module
  H+E, where H is a cumulative grade for homework and active participation and E is a grade for the exam

Bibliography

Recommended Core Bibliography

• Freidlin, Mark. Random Perturbations of Dynamical Systems / Mark Freidlin, Alexander Wentzell. –Springer, 2012

Recommended Additional Bibliography

• M. I. Freidlin, & A. D. Wentzell. (2012). Random Perturbations of Dynamical Systems (Vol. 1984). Springer.

Authors

• Kolesnikov Aleksandr Viktorovich